\begin{table}
  \centering
  \caption{The labeled poset \(\cP_X\) for a cominuscule flag variety \(X\). Each element in \(\cP_X\) is drawn as a box, and boxes decorated with an ``\(s\)'' correspond to short roots.  The partial order on boxes is given by $\alpha \leq \beta$ if and only if $\alpha$ is weakly north-west of $\beta$. This table is a modification of \cite[Table 1]{buch2022positivity}.}\label{table:poset}

  \begin{tabular}{|c|}
    \hline                                                                                                                                                    \\
    Grassmannian $\Gr(m,n)$ of type \(A_{n-1}\)                                                                                                                         \\
    \begin{dynkinDiagram}[text style/.style={scale=0.9},labels={1,2,m,n-2,n-1},scale=2]
      {A}{oo.*.oo}
    \end{dynkinDiagram}                                                               \\
    $ m \Bigg\{ \underbrace{
      \tableau{\tableausz}{
    [a]   & [a]    & [a]    & [a]                                                                                                                             \\
    [a]   & [a]    & [a]    & [a]                                                                                                                             \\
    [a]   & [a]    & [a]    & [a]
      }
    }_{n-m}$                                                                                                                                                  \\  \\
    \hline                                                                                                                                                    \\

    Odd quadric $Q^{2n-1} \subset \bP^{2n}$                                                                                                                   \\
    \multirow{2}{*}{\ \begin{dynkinDiagram}[text style/.style={scale=0.9},labels={1,2,n-1,n},scale=2]
                          {B}{*o.oo}
                        \end{dynkinDiagram}}                                             \\ \\
    $\underbrace{ \tableau{\tableausz}{
    [a]{} & [a]{}  & [a]    & [a]    & [a]    & [a]{s} & [a] & [a]{} & [a]{} & [a]{} & [a]{}
    }}_{2n-1}$                                                                                                                                                \\ \\
    \hline                                                                                                                                                    \\
    Lagrangian Grassmannian $\LG(n,2n)$                                                                                                                       \\
    \multirow{2}{*}{\ \begin{dynkinDiagram}[text style/.style={scale=0.9},labels={1,2,n-1,n},scale=2]
                          {C}{oo.o*}
                        \end{dynkinDiagram}}                                             \\ \\
    $\left. \tableau{\tableausz}{
    [a]{} & [a]{s} & [a]{s} & [a]{s} & [a]{s} & [a]{s}                                                                                                        \\
          & [a]{}  & [a]{s} & [a]{s} & [a]{s} & [a]{s}                                                                                                        \\
          &        & [a]{}  & [a]{s} & [a]{s} & [a]{s}                                                                                                        \\
          &        &        & [a]{}  & [a]{s} & [a]{s}                                                                                                        \\
          &        &        &        & [a]{}  & [a]{s}                                                                                                        \\
          &        &        &        &        & [a]{}
    } \right \} n$                                                                                                                                            \\ \\
    \hline                                                                                                                                                    \\
    Max.\ orthog.\ Grassmannian $\OG(n,2n)$                                                                                                                   \\
    \begin{dynkinDiagram}[text style/.style={scale=0.9},labels={1,2,n-2,n-1,n},label directions={,,right,,},scale=2]{D}{oo.oo*}\end{dynkinDiagram} \\
    $\left. \tableau{\tableausz}{
    [a]   & [a]    & [a]    & [a]    & [a]                                                                                                                    \\
          & [a]    & [a]    & [a]    & [a]                                                                                                                    \\
          &        & [a]    & [a]    & [a]                                                                                                                    \\
          &        &        & [a]    & [a]                                                                                                                    \\
          &        &        &        & [a]
    }\right \} n-1$                                                                                                                                           \\ \\
    \hline
  \end{tabular}
  \begin{tabular}{|c|}
    \hline                                                                                                                                          \\
    Even quadric $Q^{2n-2} \subset \bP^{2n-1}$                                                                                                      \\
    \multirow{4}{*}{\ \begin{dynkinDiagram}[text style/.style={scale=0.9},labels={1,2,n-2,n-1,n},label directions={,,right,,},scale=2]
                          {D}{*o.ooo}
                        \end{dynkinDiagram}} \\ \\ \\ \\
    $\underbrace{ \tableau{\tableausz}{
    [a] & [a] & [a] & [a] & [a]                                                                                                                     \\
        &     &     & [a] & [a] & [a] & [a] & [a]
    }}_{2n-4}$                                                                                                                                              \\ \\
    \hline                                                                                                                                          \\

    Cayley Plane $E_6/P_6$                                                                                                                          \\
    \multirow{5}{*}{\ \begin{dynkinDiagram}[text style/.style={scale=0.9},labels={1,2,3,4,5,6},scale=2]
                          {E}{ooooo*}
                        \end{dynkinDiagram}}                                 \\ \\ \\ \\ \\
    $\tableau{\tableausz}{
    [a] & [a] & [a] & [a]                                                                                                                           \\
        &     & [a] & [a] & [a] & [a]                                                                                                               \\
        &     & [a] & [a] & [a] & [a]                                                                                                               \\
        &     &     &     & [a] & [a] & [a] & [a]
    }$                                                                                                                                              \\ \\
    \hline                                                                                                                                          \\
    Freudenthal variety $E_7/P_7$                                                                                                                   \\
    \multirow{5}{*}{\ \begin{dynkinDiagram}[text style/.style={scale=0.9},labels={1,2,3,4,5,6,7},scale=2]
                          {E}{oooooo*}
                        \end{dynkinDiagram}}                                 \\ \\ \\ \\ \\
    $\tableau{\tableausz}{
    [a] & [a] & [a] & [a] & [a] & [a]                                                                                                               \\
        &     &     & [a] & [a] & [a]                                                                                                               \\
        &     &     &     & [a] & [a] & [a]                                                                                                         \\
        &     &     &     & [a] & [a] & [a] & [a] & [a]                                                                                             \\
        &     &     &     & [a] & [a] & [a] & [a] & [a]                                                                                             \\
        &     &     &     &     &     &     & [a] & [a]                                                                                             \\
        &     &     &     &     &     &     &     & [a]                                                                                             \\
        &     &     &     &     &     &     &     & [a]                                                                                             \\
        &     &     &     &     &     &     &     & [a]
    }$                                                                                                                                              \\ \\
    \hline
  \end{tabular}\label{tab:poset}
\end{table}
